[BradZax]

[You must be logged in to view images. Log in or Register.] DSM post calculator:

The \(N\)-Particle EquationFor a system of \(N\) particles (like an entire atom or molecule), the equation looks like this:

(The Second Summation)\(V_{ext}(\mathbf{r}_j, t)\): External forces acting on each particle (like a magnetic field).\(V_{int}(\mathbf{r}_j, \mathbf{r}_k)\): ,\dots ,\mathbf{r}_{N},t)=\left[\sum _{j=1}^{N}\left(-\frac{\hbar ^{2}}{2m_{j}}\nabla _{j}^{2}+V_{ext}(\mathbf{r}_{j},t)\right)+\sum _{j<k}^{N}V_{int}(\mathbf{r}_{j},\mathbf{r}_{k})\r ight]\Psi (\mathbf{r}_{1},\dots ,\mathbf{r}_{N},t)\)Breaking Down the Components1. The Energy Evolution (Left Side)\(i\hbar \frac{\partial}{\partial t}\):

This represents the change in the system over time.

\(i\) is the imaginary unit and \(\hbar \) is the reduced Planck constant.2. The Kinetic Energy (The First Summation)\(\sum_{j=1}^N -\frac{\hbar^2}{2m_j} \nabla_j^2\): This calculates the motion (kinetic energy) for every single particle in the system.

[You must be logged in to view images. Log in or Register.]

The \(\nabla ^{2}\) (Laplacian) operator accounts for movement in all three spatial dimensions (\(x, y, z\)).

R_μν = ∂_ρΓ^ρ_μν - ∂_νΓ^ρ_μρ + Γ^ρ_ρλΓ^λ_μν - Γ^ρ_μλΓ^λ_νρwhere:Γ^σ_μν = 1/2 g^σρ (∂_ν g_ρμ + ∂_μ g_ρν - ∂_ρ g_μν)G_μν + Λg_μν = (8πG/c^4) T_μν

3. The Potential Energy (The Second Summation)\(V_{ext}(\mathbf{r}_j, t)\): External forces acting on each particle (like a magnetic field).\(V_{int}(\mathbf{r}_j, \mathbf{r}_k)\):

This is what makes the equation truly "long" and complex. It accounts for the interactions between every pair of particles, such as the electrical repulsion between electrons.4.

The Wavefunction\(\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N, t)\):

This is the "cloud" of probability for the entire system. Instead of tracking one position, it tracks the coordinates of all \(N\) particles simultaneously.

(The Second Summation)\(V_{ext}(\mathbf{r}_j, t)\): External forces acting on each particle (like a magnetic field).\(V_{int}(\mathbf{r}_j, \mathbf{r}_k)\):

(The Second Summation)\(V_{ext}(\mathbf{r}_j, t)\): External forces acting on each particle (like a magnetic field).\(V_{int}(\mathbf{r}_j, \mathbf{r}_k)\):

\(\sum_{j=1}^N -\frac{\hbar^2}{2m_j} \nabla_j^2\): This calculates the motion (kinetic energy) for every single particle in the system. The \(\nabla ^{2}\) (Laplacian) operator accounts for movement in all three spatial dimensions (\(x, y, z\)).

\(\Psi(\mathbf{r}_1, \dots, \mathbf{r}_N, t)\): This is the "cloud" of probability for the entire system. Instead of tracking one position, it tracks the coordinates of all \(N\) particles simultaneously.

[You must be logged in to view images. Log in or Register.]

R_μν = ∂_ρΓ^ρ_μν - ∂_νΓ^ρ_μρ + Γ^ρ_ρλΓ^λ_μν - Γ^ρ_μλΓ^λ_νρwhere:Γ^σ_μν = 1/2 g^σρ (∂_ν g_ρμ + ∂_μ g_ρν - ∂_ρ g_μν)G_μν + Λg_μν = (8πG/c^4) T_μν

iℏ ∂/∂t Ψ(r₁, r₂, ..., rₙ, t) = [ Σᵢ₌₁ⁿ (-ℏ²/2mᵢ) ∇ᵢ² + Σᵢ<ⱼ V(rᵢ, rⱼ) + Σᵢ V_ext(rᵢ) ] Ψ(r₁, r₂, ..., rₙ, t)

The \(\nabla ^{2}\) (Laplacian) operator accounts for movement in all three spatial dimensions (\(x, y, z\)).